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Title: The Metric Matrix: Simultaneous Multidimensionality in African Music
Author: David Locke, Tufts University
Things are not what they seem. The familiar reveals an unexpected side. Shapes are
shifting. Many students of African music undergo experiences like this: an apparently
stabile musical phenomenon changes, inexplicably and by its own accord, to become
something else. In a private lesson you learn a musical phrase that you grasp well
enough, but at the next lesson it sounds very different and, when you hear the phrase in
the polyphonic context of an actual performance, you need to re-learn it once again. This
paper forwards the suggestion that many types of African music are patterned
intentionally so that this phenomenon, which I will term "simultaneous
multidimensionality," is put in motion and sustained. I will posit the "metric matrix" as a
heuristic concept that tracks patterns of accentuation and I will list features of musical
design that open for listeners the opportunity to creatively hear music of this style.
In many kinds of African music, performers set up dynamic steady states. Timed to a
steady beat, a cleverly arranged pattern of notes cycles repeatedly within a fixed span of
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time providing an opportunity for a culturally attuned listener to hear a rich set of
rhythmically exciting melodies. Composers fix multi-determinate components into their
musical designs; performers bring this multi-faceted condition into being; and listeners
actively participate in hearing the multivalent potential of a familiar item of repertory.
Examples range over vocal music, such as "pygmy" group singing, the music of tuned
instruments, such as Shona mbira music, and the music of indefinitely pitched
instruments, such as Ewe dance-drumming.1 Music like this presents to the mind's
musical ear multiple simultaneous views that are constantly in a condition of non-
resolving metamorphosis. Although works in this style often have musical forms that are
directional, narrative, and/or goal-oriented, a significant portion of their musical force
nevertheless generates from the knowing use of musical repetition that enables this
experience. Framed within the concepts of metric matrix and simultaneous
multidimensionality, this paper explores musical polysemy in Africa.2 After an
exposition of concepts, the ideas are applied to an example of dance-drumming from the
Dagomba repertory.
1 For sound recordings and discussion of these three examples, see my chapter in Worlds of Music (Locke 2009a). 2 I have applied these concepts in two recent articles. Using an item of Dagomba dance-drumming as exemplar, I compare the idea of simultaneity in African music with its use in cubist visual art (Locke 2009b); I use the concept of the metric matrix in a descriptive analysis of Yewevu, an Ewe musical work (Locke, under review).
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PART ONE: IDEAS
Syntax
When practitioners of this style of African music compose and perform, one aesthetic
goal that has impact upon the design of their musical choices is to establish and sustain
the music's open-ended, iridescent quality. Guided by inherited traditions that constrain
utterly free choice, musicians achieve their expressive intent by working within the
framework of a musical syntax that is well established but largely un-verbalized within
their cultural community. Although enormously varied across Africa and its diaspora in
particular detail according to such factors as ethnic heritage, the history of inter-cultural
contact and genre-specific musical features, this paper argues for the presence in Africa
of a shared musical syntax that produces music with the aesthetic quality of everlasting
energetic vitality.3
My assertion about the presence of syntax runs counter to what Simha Arom has written
about Central African music.
3 I subscribe to the set of aesthetic values for African traditional performance arts listed by Robert Farris Thompson, which includes this quality of life-force energy (Thompson 1974).
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All the possible realizations of a given rhythmic figure are culturally speaking
identical so that the order in which they are repeated is almost always random. . .
. [T]he order in which the realizations of a given figure are concatenated is
optional. This in turn means that no syntactic constraints apply. (Arom 1991 p.
299)
In my view, Arom's denial of syntax implies that African musicians are, so to speak,
mindlessly pulling phrases out of a hat. On the contrary, I argue that musical syntax is
activated during performance purposely to achieve the aesthetic goal of keeping the
music's polyphonic texture in a constant state of process. African musicians, I suggest,
intend to impart a multivalent quality to notes, figures, motives, and phrases.
Two Facets for Analysis
Analysis of an example of African polyphonic music may emphasize its parts or its
totality. On the one hand, African ensemble music for dance may be understood as a
counterpoint among separate phrases, which Robert Farris Thompson has dubbed "apart
playing" (Thompson 1966 p. 92). In this approach, analysis considers the linear design of
each phrase and the network of connections between and among phrases. This paper
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shows that the patterning of "phenomenal accents" in phrases can be explained in terms
of their rhythmic relationship to an implicit matrix of steady beats (Lerdahl and
Jackendoff 1983, p. 17). In some cases, recurring accents consistently coincide with
beats or else suggest their displacement; in other cases, accents that shift between onbeat
and offbeat locations create temporary feelings of musical motility and stability.4
In the second approach, the ensemble's music may be heard as a well-blended whole
whose melodies arise from the careful arrangement of interlocking phrases, which Meki
Nzewi has named the "ensemble thematic cycle" or ETC (Nzewi 1997 p. 42).5
Composers and performers have in mind composite melodies that listeners hear by
mentally conjoining tones from the several parts. Often, the ETC is an instrumental
setting of texts in the local language that are connected to the occasion of performance
within community. The temporal frame of the ETC is a recurring time span whose
internal structure has impact upon the perception and expressiveness of sounded music.
In most cases, musical ethnographic research reveals that the ETC has a sense of
4 Patterns of onbeat and offbeat accentuation are elaborated in Locke 1982. 5 The ETC idea differs from polyrhythm in its acceptance of melody and harmony as intentional features of African polyphony. ETC also permits inclusion as a core musical feature the latent texts that underlie many idioms of African instrumental performance, rather than marginalizing language as extra-musical.
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rhythmic motion towards a cadence moment of temporary stasis and resolution. Just as
listeners can aesthetically enjoy the separate phrases or the polyphony of the ETC,
analysts of African polyphonic music at steady tempo should attend both to the whole
and the parts of which it is made.
The Metric Matrix
In African music of the kind discussed here, sounded music happens in the context of an
unsounded temporal structure that I dub the "metric matrix."6 Beats, which may be said
to contain sets of pulses or time-points, are the factors within the matrix most present to
consciousness. Meter is usefully regarded as a matrix of beats of different duration and
position within an isochronous time span that recycles repeatedly during performance.
Beats flow at steady tempo, shaping musical time into equidurational units that are
available to listeners and players. For listeners, the choices within the metric matrix are a
means for orientation to the gestalt of the ETC; for composers and performers, the
positions of individual beats and/or the durational values of a series of beats guide
rhythmic creative choice.
6 The meter-as-matrix concept has been a feature of my writings about African music since my doctoral dissertation. For monographs that use this idea see Locke 1988, 1990 and 1992.
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Two important types of metric matrix operate in most idioms of African dance music: (1)
ternary time, i.e., beats contain three fast pulses, and (2) quaternary, i.e., beats contain
four fast pulses (Figure 1 and Figure 2). ETC time spans usually are multiples of two.
When I visualize in staff notation African "groove" music that has never before been
graphically represented, I inscribe the cadential moment of the ETC on the first onbeat
time-point in the measure.
Figure 1 Metric matrix in ternary-quadruple time
Figure 2 Metric Matrix in quaternary-quadruple time
Each moment within the metric matrix has an inherent rhythmic valence.7 The onbeat
position of beats defines their primary temporal feeling. Although each onbeat accent
within the time span of the ETC usually is articulated with equal loudness, position order
in the metric set of beats does have rhythmic significance. In the concept argued here,
the musical rhythm of separate phrases and the entire ETC is shaped by motion towards a
7 Arguing in favor of merging the two concepts, Christopher Hasty reviews the historical bifurcation of meter and rhythm that has characterized music theory studies of Western art music (see Hasty 1997, p. 3-21). The concepts forwarded here, developed independently of Hasty's work, are consonant with his approach.
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commonly felt downbeató"the one" in the parlance of non-African students of African
music.
In his analytic writing, J.H.K. Nketia asserts the idea of "regulative beat," that is, a
steadily felt and/or articulated unit of duration (Nketia 1963, p. 64). Willie Anku, while
accepting the presence of beats and time-points within them, finds them to serve a
"structural" rather than "metric" purpose (Anku 1995, p.177). With regard to rhythmic
motion, Anku's regulative time-point, or RTP, is the recurring time-point on which a
phrase begins in each pass through its time span. As mentioned above, in transcription I
intentionally set the cadential time-point of the overall ETC on the first beat of measures,
which enables me to call it the downbeat, thus purposely invoking a metrical/rhythmical
value. In my experience, the total number of time-points within the ETC--what Anku
calls "a time-point set"--is not actively used by African musicians (Anku 2000, paragraph
9). Instead, musicians feel beats in relation to sounded patterns such as a time-keeping
ostinato, or "time line" in Nketia's widely adopted terminology (Nketia 1963, p. 78).8 To
identify time-points within time spans, I use the following numbering system for ternary-
duple time: 1.1, 1.2, 1.3, 2.1, 2.2, 2.3. Time-points marked 1.5 and 2.5 are binary
upbeats, that is, the mid-point between successive onbeats.
8 Permit an anecdote from my field work in Ghana (1975-1977). Frustrated with my painstaking effort to discern the relationship of strokes in a lead drum phrase to notes in the time line, my teacher Godwin Agbeli remarked, "That way will take forever. You should use the bell to find the beats. Then feel the drum phrase in relation to the beats" (paraphrase, personal communication).
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African musicians creatively manipulate onbeats in two ways: displacement and tactus
change. In displacement, beats may be shifted from their onbeat positions to the several
offbeat time-points contained within the beat's duration. In tactus change, the rate at
which beats are present to musical attention varies by factors of two, a process akin to
"double time" or "cut time" in African diasporic music like jazz.
Three-With-Two
Serving as a creative resource for composition and improvisation, three-in-the-time-of-
two (3:2) pervades African music with beats of ternary structure. In the seminal words of
A.M. Jones,
We have to grasp the fact that if from childhood you are brought up to regard
beating 3 against 2 as being just as normal as beating in synchrony, then you
develop a two-dimensional attitude to rhythm . . . . This bi-podal conception is . .
. part of the African's nature . . . . (Jones 1959, p.102)
In my view, the musical condition is best characterized not as 3 against 2 but rather as 3
with 2, i.e., the simultaneous presence of both ways to organize perception of musical
time. In other words, ternary beats imply their binary/quaternary counterparts; 3:2 is an
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inseparable twinning of two complementary feelings of musical time.9 Whereas African
musicians, who play in a "writing-free" tradition, need not rank one time-feel as
fundamental, we analysts of African music, working in a "writing-constrained" condition,
are pressed to make this choice. In my view, ample ethnographic musical evidence
suggests the most culturally appropriate figure-ground hierarchy in ternary-duple music:
the "two feel" may be taken as the foundation in terms of which the "three feel" gains its
effect, a condition replicated in ternary-quadruple music between the "four feel" and "six
feel."10 While 3:2 pervades ternary music, quaternary music seldom uses tuplets; instead,
a set of dotted notes may temporarily make 3:2 and 3:4 temporal structures.
Rhythms using 3:2 relationships need not start at the moment of unison between the two
beat flows. In other words, 3:2 and 3:4 can be phrased in different ways. According to
my experience, the artful beauty of phrases with 3:2 patterning frequently draws upon
motion towards the resolving, cadential moment with the two streams temporarily
9 Unlike Justin London, this paper suggests that in African music, at least, polymeter is possible and is very likely to exist: one phenomenal musical object can be felt in multiple ways simultaneously (see London 2004, p. 50). 10 Steve Friedson, arguing for what he calls a "strong position of polymeter," asserts that both time feels are "copresent at all times" (Friedson 2009, p. 208).
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coincide. The "phrasings of 3:2," so to speak, are shown in Figure 3 and Figure 4. (see
Locke 1982, p. 236).
Figure 3 Three phrasings of 3:2
Figure 4 Two phrasings of 2:3
Rhythmic Accentuation in the Metric Matrix
Thinking of rhythm, tones in a phrase that coincide with an implicit beat acquire the
distinctive quality of "onbeatness." Beat one in each series connotes a feeling of arrival
and temporary stasis that justifies the label "downbeat." Each beat within the ETC has
unique character; for example, beats in a four-element set tend to be graded 1-3-4-2 in
terms of their capability of being at rest (stabile) or in motion (motile). Thus, a tone that
occurs on beat two will inherently feel more motile, that is, least resolved and most
capable of being set in motion, than a tone whose onset matches beat one. At the faster
durations of time-points within beats, the feelingful quality of each offbeat moment also
guides musical decision-making and has impact upon a listener's affective response. The
rhythmic effect of notes varies with their metric positions; terms that I use to identify
rhythmic qualities of offbeat time-points include:
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-Before-beat: emphasis on offbeat position just prior to a subsequent onbeat.
-After-beat: emphasis on offbeat position just after to a prior onbeat.
-Upbeat: emphasis on offbeat position at mid-point between two adjacent beats.
-Displaced beat: felt as a new onbeat location for a beat of identical duration.
-Pickup: rhythmically associated with subsequent onbeat towards which it moves.
Since there are many beat streams in the implicit metric matrix, however, the rhythm of a
sounded phrase is subject to cognitive re-orientation depending on the flow of beats on
which perception is grounded. 11 Within the enigmatic and paradoxical context of the
metric matrix, tones are onbeat and offbeat simultaneously.
Simultaneous Multidimensionality
"Simultaneous multidimensionality" names a condition in which music is coherent from
several perspectives at the same time.12 Repetition is the crucial force that enables the
11 Derived from my doctoral dissertation, my first paper on musical phenomena like cognitive re-orientation and the relationship between rhythmic accentuation and metric structure may be found in Ethnomusicology (Locke 1982, pp. 224, 227-243). 12 For me, the seminal paper on this phenomenon in African music is Gerhard Kubik's 1962 article, "The Phenomenon of Inherent Rhythms in East and Central African Instrumental Music."
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music to achieve this crystalline depth.13 In performance, the composite whole of the
ETC circles around its music axis, enabling a creative listener to contemplate the
polyphony as always renewing itself, always creating its musical identity. Because of the
cyclic nature of time in this musical style, any element not only points forward towards
what comes next but also responds to what has come before. Listeners experience each
pass through the ETC in relation to prior and subsequent musical action. Who can find
the circle's beginning or end? Music of this design achieves sculptural persistence when
heard by listeners who are attuned to simultaneous multidimensionality.
Devices of simultaneous multidimensionality include:
(1) Dualism of tempo: feeling the flow of time at different rates; not only according to 3:2
relationships but also in ratios of augmentation and diminution, as in double time or cut
time.
(2) Meter as a matrix: changing the figure-ground relationship between notes in a
sounded phrase and an implicit grid of beats.
13 "Repetition" should not be understood as a recurrence of exactly the same thing, for that is impossible. Even if the phrases in the ETC could be perfectly repeated, they would be in the existential condition of items in a series. Devices of creative hearing and phenomenal variation further challenge the fallacious notion of sameness.
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(3) Polyphonic perception: perceiving a multiplicity of composite melodies that emerge
from the many ways separate phrases may be heard in counterpoint.
(4) Phrase reconfiguration: hearing a phrase as starting, moving and ending in different
ways as the music recurs again and again.
(5) Equivocal phrase shape: in some cases, durational values and/or pitches within a
phrase do not make a clear design, thus requiring the listener to project a temporal shape
onto the heard aural image.
(6) Polysemous phrases: in some cases, durational values and/or pitches within a phrase
make a clear design that accentuates several different beat streams equally well.
The perceptual conditions that enable this plural mind-body cognitive condition are
particularly likely to arise in ternary time with music whose phrases are structured within
a 3:2 (three-in-the-time-of-two) temporal framework. In quaternary time, simultaneous
multidimensionality arises more from (1) the dualism between onbeats and upbeats, (2)
1:2 shifts in tactus, (3) constant accentuation of the second or fourth elements in a four
item set, and (4) changing perception of phrase shape.
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Given limits on the scope of this paper, I will now discuss one example in ternary time to
exemplify the concepts presented above.
PART TWO: EXAMPLE
Dagomba Dance-Drumming: "J—rigu N-dari O Salima"
"J—rigu N-dari O Salima" may be regarded as a "piece" of music in a genre of dance
music called Praise Name Drumming, or "salma" in the local Dagbani language (literally,
"story-telling") of the Dagomba people from northern Ghana.14 It salutes the generosity
of spirit of the late Kar-naa Ziblim, an important regional sub-chief of the polity called
Kariga, by announcing in drum language, "A foolish man tries to buy our respect with
money, but a wise man earns our respect by acts of good character." 15 "J—rigu" is music
for solo dancing: at events such as festivals and life-cycle celebrations descendants of
Kar'naa Ziblim may request that drummers play it for them when they dance in the public
plaza. The cognoscenti in the audience will realize the dancer's genealogy and may
ponder the relevance of the proverb in today's world.
14 For extensive information about Dagomba dance-drumming, go to http://dagomba.uit.tufts.edu. (See also Locke 1990 and publications by John Chernoff.) 15 In the Dagomba hierarchical political system, the chief of Kariga is at the second rank below the Yaa Naa, or king (see Staniland).
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Two types of drums make the instrumental music. Luŋa, a two-headed hourglass-shaped
pressure drum, plays melodies of three primary pitches--low, mid, and high, represented
here on the 5-line staff as B-D-E--by varying the tension on the drum heads; notated with
tied notes, sliding effects to and from unaccented auxiliary pitches below and above the
accented primary pitches are crucial to making the drum sound like the spoken Dagbani
language. Guŋ-g„ŋ, a two-headed cylindrical drum, plays loud accented tones of two
timbres, bounce and press; this discussion will omit the guŋ-g„ŋ's quiet unaccented
"filler" tones. A solo vocalist, who sings proverb-laden lines that allude to the life and
times of person being saluted, also is part of the ensemble. In a performance that might
last for several hours, the musical ensemble will "work the crowd" by inviting members
of the audience out to dance one-by-one to different pieces of Praise Name Drumming.
Well-wishers shower the dancers with money that goes to the musicians.
The material presented here emerged from my study with the late Alhaji Abubakari
Lunna from 1975-2009. Alhaji Abubakari and I worked on lead drumming in Dagomba
dance-drumming music using the inherited tradition of pre-composed phrases, which
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Alhaji would term "talks" (in Dagbani "baŋsem" or "knowledge"). Saying that he was
following standard Dagomba pedagogical method, Alhaji taught a musical work by
giving me a set of "talks" that an informed Dagomba listener would expect to hear when
that piece is played. A "talk" is a drummed theme consisting of one or more short
phrases. Shaped by the Dagbani text it sets, each theme has a distinctive musical design.
In performance, after the lead drummer calls in the ensemble, the "talks" are played in a
sequence likely to make compelling danceable music.16
The time-span of "J—rigu's" ETC is eight ternary beats, which I variously notate in
measures of two, four, and eight beats depending on the purpose of the transcription.
Shown here in duple bars with one-line staffs that emphasize rhythm and form, the call-
and-response is a straightforward alternation (see Figure 5).17
Figure 5 "J—rigu" call-and-response form
16 This situation is reminiscent of the case mentioned by Arom, above. The "talks" can indeed be played in any order. My quarrel with Arom is his failure to convey that musicians do understand the musical impact of the phrases they play, that they can vary the phrases to achieve musical goals, and that their choice of phrase can be influenced by their understanding of musical syntax. 17 This figure in ternary-duple meter is borrowed from the website that presents many different items of repertory that have a variety of ETC time spans (. To facilitate comparison, all pieces are transcribed in duple bars.
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The lead luŋa drum has several preset themes, three of which are shown here (see Figure
6). Alhaji Abubakari taught theme 1 of the solo lead luŋa drum, which means "foolish
man buys gold," as the call that requests response from a group of answer luŋa drums and
two guŋ-g„ŋ drums, which render the same text--"wise man buys respect"--with slightly
different musical phrases (see Figure 7 and Figure 8)
Figure 6 "J—rigu": lead luŋa themes 1, 2, 3
Figure 7 "J—rigu": answer luŋa phrase
Figure 8 "J—rigu": guŋ-g„ŋ phrase
Transcribed in ternary-quadruple time, theme 1 of the lead luŋa opens with a dramatically
offbeat high-pitch tone on the third time-point within beat one. Introduced by a pickup
on the third time-point in beat two, the theme follows with four quarter notes of gradually
rising pitch, L L M M. The accented notes, which show when the stick strikes the drum
skin, are in 3:2 relationship to the implicit ternary beats. Receiving this gesture from the
lead luŋa, the answer luŋa plays six notes that continue the 3̀:2 temporal relationship
between the sounded accents and the implicit ternary metric structure while undulating
downward from mid- to low-pitch--M M M H L L. The response drum continues the
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temporal values and 3:2 structure of the lead drum but changes the phrasing of 3:2.
Whereas the lead drum's 3:2 phrasing goes 1 2 3 1, the answer luŋa phrases 3:2 as 2 3 1,
2 3 1. This change in phrasing achieves the musical effect that I am calling an "open-
ended feeling" to the flow of time. When the lead luŋa re-enters on the next pass through
the ETC, it's first note becomes multidimensional: it can be heard as continuing the
answer luŋa's flow of quarter notes, but because its first note is three eighth notes in
duration, it also can be heard to boldly accentuate time-point 1.3, thus interrupting the 3:2
flow and bringing attention to this offbeat location. Although third time-point positions
within beats are articulated by the quiet pickup note in beat two and the strongly
articulated stroke in beat three, theme 1 does not forcefully suggest displacement of the
ternary onbeat.
Like leading luŋa, guŋ-g„ŋ has a polysemous pattern of accentuation. It readily suggests
"in four" accentuation of ternary beats 2 3 4 1 if its short accented notes are felt as pick-
ups to the subsequent onbeat accents, "a 2, 3 a 4, 1." But since the short strokes are
supposed to be strongly struck, the rhythm of the phrase can be heard "in six," "2 &, 4 5
&, 1" (see Figure 9).
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Figure 9 "J—rigu": guŋ-g„ŋ phrase "in four" and "in six"
In my experience, the guŋ-g„ŋ's contribution to the multidimensionality of the ETC is to
accentuate the four-feel in contrast to the six-feel so strongly articulated by the answer
luŋa, while also adding support to the rhythmic multideterminancy of the lead luŋa part.
Taken in tandem, the rhythmic composite of the accents in the answer luŋa and guŋ-g„ŋ
phrases sounds out two sets of 3:2 phrased 2 3 1 2 3 1 (Figure 10).
Figure 10 "J—rigu": rhythmic composite of answer luŋa and guŋ-g„ŋ
Thus far the analysis has shown that multideterminancy in time-feel and phrasing is
structurally built into the composition. Now we consider how variability in the design of
lead drum phrases correlates to temporal positions within the metric matrix. The metric
matrix provides a useful analytic tool with which to articulate the musical logic of these
phrases. Let us look at three themes for "J—riga." The meaning of theme 1 has already
been discussed. Theme 2 points to the limitations of money, suggesting that personal
integrity can never be used up. Theme 3 simply announces that Kar-naa Ziblim has
become "the lion," i.e., the top-ranked chief in the Kariga estate.
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Theme 1 J—rigu n-dari o salima. Foolish man buys gold. j—rigu foolish man n-dari he buys o it salima gold Theme 2 Ŋunda bi yo ku landa. Debtor cannot buy more. ŋunda the person who buys bi yo never pay ku landa cannot buy more Theme 3 Gbungbiri l—li m-bala la. This is the lion's place gbugbiligo lion l—li place m-bala la this is
The following figure shows an eight-beat matrix of ternary and binary beats in onbeat and
offbeat positions (see Figure 11). By scanning vertically, we can observe the coincidence
between notes in a lead luŋa theme and these marks in the matrix. For example, in theme
1 we observe coincidence with binary beats four to seven and the third offbeat time-
points in ternary beats one, two and three.
Figure 11 Metric matrix (ternary time) and "J—rigu" lead luŋa themes 1, 2, 3
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Theme 2 begins with four drum strokes that coincide with the upbeats of binary beats one
to four; the fourth note, which is short, is followed by three drum strokes that coincide
with the binary onbeats five, six and seven. In other words, the seven accents in theme 2
may be heard as two sets of three long notes that are stitched together with one short note
that flips the accents from the upbeat to the onbeat of the stream of binary beats in the
metric matrix. As discussed below, the lead luŋa player usually "marks time" with a fill
phrase during the response from the answer luŋa and guŋ-g„ŋ (see Figure 12). The "fill
phrase," timed to the binary upbeats, phrased 1 2 3 1 2 starting on the sixth ternary
onbeat, flows smoothly into the opening gesture of theme 2. 18 The rhythm of theme 2
manifests two phrasings of 3̀:2: strokes one, two and three move 3 1 2, while strokes five,
six and seven move 2 3 1. Theme 2 starts on the second time-point in the eight-beat
cycle rather than the third, which is the point that launches themes 1 and 3. Since three of
its strokes coincide with the second offbeat time-point in ternary beats one, three and
18 Coinciding with even-numbered ternary onbeats, the binary upbeat accentuates the music's "backbeat." Since the final upbeat binary beat in the cycle strikes time-point 8.3, when inserted into theme 1 the "fill" motive adds its weight to the theme's subtle accentuation of third ternary offbeats.
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four, this position in the metric matrix may be identified as an aspect of the musical
personality of phrase 2.
Theme 3 uses a new durational value--the double time beat in ternary time, notated here
as dotted eighth notes. Theme 3 powerfully conveys a sense of urgent, quickened motion
from ternary beat two towards cadence on ternary beat five, the mid-point of the cycle
and focal point of the response drum parts. These time values create two different
positions of a 4:3 temporal structure with the binary onbeats and upbeats. Given a
sequence of six notes of equal duration, the drummer may shape the equivocal rhythm
with dynamic accent to the offbeat notes.
This exposition demonstrates the heuristic value of the metric matrix for analysis of the
rhythmic design of phrases. The analysis has shown that the music of the lead drum
injects a variety of rhythmic effects into the ETC, thus maintaining the overall aesthetic
value on "a process of becoming." By injecting diverse ways to perceive both the gestalt
of the ETC as well as the rhythmic meaning of particular notes and phrases, the lead
drum part maintains "J—riga's" musical multidimensionality.
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As mentioned above, analysis of this kind of African music does well to include whole
textures as well as individual phrases. What of relationships among sounded phrases in
the ETC? Let us consider theme 2 (see Figure 12).
Figure 12 "J—rigu": lead luŋa theme 2 in metric matrix and ETC
The musical example shows the ternary onbeats, and the onbeat and upbeat positions of
the 3̀:2 binary beats. Lead drum theme 2 is given in vocables, as well as the Dagbani
drum language. The "fill" phrase, which sets no language, usually is lightly struck on
very low pitches about a perfect fourth or fifth below the low-pitch tone. It is important
to observe that the "fill", like the theme itself, is timed to the upbeat location of the binary
beats. We further see that this creates 3:2 structures, phrased 1 2 3, 1 2 3, with the
ternary onbeats; moments of coincidence are on ternary beats six, eight and two, the
"backbeats" of the music's main onbeat ternary groove. The "fill" phrase of the lead luŋa
interlocks with the accents of answer luŋa to make a quick back-and-forth texture whose
increased density intensifies the music's texture, making it feel that the pace has become
faster. This interlock certainly reinforces the music's overall sense of
multidimensionality. In duet with guŋ-g„ŋ, theme 2 of lead luŋa reinforces the ternary
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onbeat interpretation of its phrase--kaKA, KA kaKA, KA. I forward this interpretation
because the onbeat notes in the lead luŋa's "fill" phrase match directly with the strokes
two and five of guŋ-g„ŋ, which coincide with ternary onbeats six and eight.
CONCLUSION
This paper suggests a way to find order in the compositional design and performance
decisions of African music with steady beat. It assumes the presence of the aesthetic goal
of simultaneous multidimensionality, that is, the creation of a musical surface that can be
heard from multiple perspectives at the same time. It asserts that this goal is achieved
through systematic means, i.e., through the workings of a musical syntax. The paper
proposes features of this syntax, especially the idea of meter as a matrix of beats.
Ordered within sets, beats have intrinsic rhythmic values, including motility and stasis.
Beats in the matrix cay be displaced to offbeat locations; the pace at which beats are
perceived can be expanded or compressed. The paper explains the patterning of sounded
musical events in terms of coincidence with the implicit moments in the metric matrix.
Three-in-the-time-of-two, phrased in several ways and occurring at different durational
26
values, is taken as a core musical structure. Repetition of phrases in a polyphonic texture
is taken as an essential enabling factor for simultaneous multidimensionality.
27
REFERENCES CITED
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Cambridge, MA: MIT Press.
28
Locke, David. 1982. "Principles of Offbeat Timing and Cross-Rhythm in Southern Ewe
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four-feelonbeat
four-feeldisplaced 1
four-feeldisplaced 2
six-feelonbeat
six-feelupbeat
three-feelonbeat
three-feeldisplaced 1
three-feeldisplaced 2
three-feeldisplaced 3
eight-feelonbeat
four-feelonbeat
four-feelupbeat
eight-feelonbeat
eight-feelupbeat
two-feel
two-feelupbeats
"two"
"three"
composite
1
A. 3:2 (three-in-the-time-of-two, "three with two") phrased 1 2 3
2
1
2
repeat continuously
1
2
3
1
2
3
start
start
"two"
"three"
composite
1
B. 3:2 (three-in-the-time-of-two, "three with two") phrased 2 3 1
2
1
2
repeat continuously
1
2
3
1
2
3
start
start
"two"
"three"
composite
1
C. 3:2 (three-in-the-time-of-two, "three with two") phrased 3 1 2
2
1
2
repeat continuously
1
2
3
1
2
3
start
start
"three"
"two"
composite
1
A. 2:3 (two-in-the-time-of-three, "two with three") phrased 1 2
2
3
1
2
repeat continuously
3
1
2
1
2
start
start
"three"
"two"
composite
1
B. 2:3 (two-in-the-time-of-three, "two with three") phrased 2 1
2
3
1
2
repeat continuously
3
1
2
1
2
start
start
q. = 1202 3
4
lead luŋa
answer luŋa
guŋ-g„ŋ
J—rigu N-dari O Salima
q. = 120
2
J—rigu N-dari O Salima lead luŋa theme 1 "J—rigu n-dari o salima"
J—ri
-gu n-dari
o
sali
ma
-
1q. = 120
2
J—rigu N-dari O Salima lead luŋa theme 2 "Ŋunda bi yo ku landa"
Ŋun
da- bi
yo ku
lan -da
1q. = 120
2
J—rigu N-dari O Salima lead luŋa theme 3 "Gbunbiri l—li m-bala la"
Gbun biri- l—
li- m-ba
la- la
q. = 120
J—rigu N-dari O Salima answer luŋa
Y—n
dan'- n-dar'
o jil'
-ma
q. = 120 2
J—rigu N-dari O Salima guŋ-g„ŋ
ka ka
ti ka
ka ka
ti ka
zi zi
gi zi zi
gi zi zi
gi zi
1
2
zi
ka ka
ti ka
ka ka
ti ka
zi zi
gi zi zi
gi zi zi
gi zi
q. = 120
beatsternary/quadruple
guŋ-g„ŋin four
guŋ-g„ŋin six
beatsbinary/sextuple
1
2
Composite rhythm
answer luŋa
guŋ-g„ŋ
2 &
3
1
2 &
3
1
q. = 120
onbeatsternary
onbeatsbinary
upbeatsbinary beats
onbeats - upbeatsternary double-time
ternary offbeatssecond time-points
ternary offbeatsthird time-point
lead luŋatheme 1"J—rigu"
lead luŋatheme 2
"Ŋunda"
lead luŋatheme 3
"Gbunbiri
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
J—ri
-gu n-dari
o
sali
ma
-
Ŋun
da
- bi
yo ku
lan -da
To gbun
biri- l—
li- m-ba
la- la
q. = 120
beatsternary
3:2 beats"onbeat"
3:2 beats"upbeat"
lead luŋatheme 2
"Ŋunda bi yo"
answer luŋa
guŋ-g„ŋ
1
time span begins
2
3
4
5
6
7
8
1
time span renews
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
dahanŊun
drum talk
dayanda
-denbi
deyo
diyanku
dahanlan
dayanda
ta
"fill"
ta
ta
ta
ta
dahanŊun
-
denY—n
dehendan
n-dariden
diyano
dahanjil'
dayanma
-
ka ka
ka
ka ka
ka
